__________________________________________________________________________________
Calculus I Final
Exam December
10, 2002
Name____________________ R. Hammack Score
______
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(1) Evaluate the following limits.
(a)
(b)
(c)
(d)
(e)
(f)
(2) Use the limit
definition of the derivative to find the derivative of the function .
(3) Find the following derivatives.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(4) Simplify the following expressions
as much as possible.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(5) Sketch the graph
of
(6) Solve the equation for
x.
(7) The questions on this page concern
the one-to-one function
(a)
(b)
(c) List the vertical asymptotes
of f (if any).
(d) List the
horizontal asymptotes of f (if
any).
(e) Find the inverse of
f.
(f) State the domain of f. All
values of x except 2.
(g) State the domain of.
(h) State the range of f.
(i) State the range
of.
(j) Find the equation of the tangent
line to the graph of at
the point where .
(8) This problem concerns the function
.
(a)
(b)
(c) Find the interval(s)
on which f is
increasing and on which it is decreasing.
(d) Find the interval(s)
on which f is
concave up and on which it is concave down.
(e) List the x-coordinates
of all inflection points of f.
(f) List all the critical
numbers of f.
(g) List the x-coordinates
of the relative extrema of f (if
any) and say whether there is a relative minima or a relative maxima.
(9) Given the equation , find .
(10) A rectangular box with with two
square sides and an open top is to have a volume of 36 cubic feet. Find the
dimensions of the container with minimum surface area.
(11) A spherical balloon is deflated
in such a way that its radius decreases at a constant rate of 15 cm/min. At
what rate is air escaping when the radius is 2 cm?
Hint: The volume of a sphere of radius r is .